Higher-order Van Hove singularity in magic-angle twisted trilayer graphene

Twisted trilayer graphene (TTG) has recently emerged experimentally as a fascinating playground to study correlated and exotic superconducting phases. We have found that TTG hosts a zero-energy higher-order Van Hove singularity with an exponent $-1/3$ that is stronger than the one predicted in twisted bilayer graphene. This singularity is protected by a threefold rotation symmetry and a combined mirror-particle-hole symmetry and can be tuned with only the twist angle and a perpendicular electric field. It arises from the combined merging of Van Hove singularities and Dirac cones at zero energy, a scheme that goes beyond the recent classifications of Van Hove singularities in single-band models. This structure gives a topological Lifshitz transition, with anomalous exponent $-2/5$, which can be achieved in TTG by varying a third control parameter such as the atomic corrugation. The interplay between the nonstandard class of higher-order Van Hove singularities and interaction effects offers an unprecedented platform for studying correlation and superconductivity.

Figure 1

(a) DOS as a function of the displacement field $D/{\epsilon }_0$ and of the number of electrons per moiré unit cell for $r=0.8$ and $\theta =1.{59}^{\circ }$. The location of the two VHS is indicated by red dots. (b) Line cuts, corresponding to the white lines in (a), of the DOS as a function of energy for various displacement fields. (c) Evolution as a function of the displacement field of the saddle points (red lines) yielding the strongest peak in the density of states. Arrows indicate increasing $D/{\epsilon }_0$. (d) Values of the parameters of the Hamiltonian (2) $H_{\mathbf{K}}\left(\mathbf{q}\right)$ as a function of $D/{\epsilon }_0$.

Figure 2

(a) Three-dimensional trajectory of higher-order VHS originating from the first magic angle ${\theta }_{CH}\simeq 1.{536}^{\circ }$ at $r=0$ and $D=0$. The blue dot gives the position of the Lifshitz transition with exponent $-2/5$. (b) Evolution of the DOS prefactor in Eq. (4) as a function of $2\gamma /\xi$. (c) Evolution of $2\gamma$ and $\xi$ as a function of $r$ along the trajectory of higher-order VHS. The inset shows the ratio $2\gamma /\xi$ in logarithmic scale.

Figure 3

(a)–(c) The positive isoenergy contours around $\mathbf{K}$ at the higher-order VHS for $2\gamma /\xi =0.1$, $2\gamma /\xi =1.0$ (Lifshitz transition) and $2\gamma /\xi =20.0$, respectively. Red lines mark the asymptotes of open orbits. (d) Landau level spectrum as a function of $2\gamma /\xi$ for $B=0.01\text{T}$. The vertical red line indicates the location of the Lifshitz transition. The Landau level energy is measured in units of $\xi /l_B^3$, where $l_B$ is the magnetic length $l_B=\sqrt{\hslash /eB}$.